Overview

Among the multitude of mathematical tools to conquer integral computations, the u-substitution is one of the most fundamental, as it helps simplify otherwise ugly-looking integrals. Let's get familiar with them in this handout!

1. Definition and Examples

1.1 Why u-sub Works

Definition: This integration technique is basically applying reverse chain rule by the Fundamental Theorem of Calculus. It aims to write integrals into the form

which, when using a substitution of $u = g(x)$, gives:

because $g'(u) = \dfrac{du}{dx}$.

1.2 Steps to Perform a u-sub

There are a few steps to performing a u-substitution. As an example, let's say we wanted to evaluate

Step 1 — Choose a simplifying transformation. Here, notice that if we could apply the power rule, that would be wonderful. Hence we choose the substitution

Step 2 — Determine the corresponding differential. We want to change everything from the $x$ world into the $u$ world. Computing $\dfrac{du}{dx}$ by differentiation:

Step 3 — Change the bounds of integration. This part only applies if you have definite integrals. For $x = 3$: $u = 1 + 69(3) = 208$. For $x = 5$: $u = 346$.

Step 4 — Assemble the integral. Putting all three steps together:

From here, it is straightforward to perform the integral because we know the integral of $\sin(x)$.

The reason we perform all the steps above is because u-substitution is basically the same thing as changing a graph's axes — if one pictures integrals as areas under curves, then u-substitution is mapping one $x$-axis into another $u$-axis. Not only do the scales on the axes change (which implies that $dx$ changes into $du$), but also the bounds of integration.

Note that you don't always have to perform substitutions of $u$ as a function of $x$. You could also perform substitutions where $x$ is a function of $u$, as we will see below.

1.3 Trigonometric Substitutions

As you continue on your calculus journey, you will encounter more and more substitutions. One particular family of substitutions are trigonometric substitutions. The reason they are so useful — and even deserve to be named "a family of substitutions" — is because of the trigonometric identities, which can help simplify an integral greatly.

Example 1: Let's consider the integral:

At first glance, this seems hard because you can't do reverse-power-rule on this. However, there is a beautiful coincidence:

The derivative of $\tan x$ is exactly the same as $1$ plus itself squared! This motivates the substitution $x = \tan u$, which implies $dx = \sec^2 u\,du$. Plugging in:

The original integral then evaluates to

which one can further simplify using trigonometry.

The key insight is that whenever you see a certain form of function in the integrand, one should use a certain substitution. The problem then transforms from a polynomial-type function into a trigonometric-type function.

Here is a list of useful trigonometric substitutions:

Form	Substitution	Reason
$\sqrt{a^2 + x^2}$	$u = a\tan x$	$1 + \tan^2 x = \sec^2 x = (\tan x)'$
$\sqrt{a^2 - x^2}$	$u = a\sin x$	$1 - \sin^2 x = \cos^2 x = ((\sin x)')^2$
$\sqrt{x^2 - a^2}$	$u = a\sec x$	$\sec^2 x - 1 = \tan^2 x = \left(\frac{(\sec x)'}{\sec x}\right)^2$

where $(f(x))' = \dfrac{d}{dx}f(x)$ and $(\sec x)' = \sec x \tan x$.

1.4 A Note on Substitutions

As seen above, the list of substitutions is extensive — there are many types out there, including Weierstrass substitutions, Euler substitutions, and many other tricks one will discover along their path through calculus.

The key to substitutions is not memorising all of them, but rather choosing the correct one which simplifies the problem at hand.

2. Problems & Practices

Problem 1: Evaluate the integral

Problem 2: Evaluate the integral

Problem 3: Evaluate the integral

Problem 4: Prove the King Property:

Problem 5: Evaluate the integral

Hint: use the result from Problem 4, and think about the periodicity of the sinusoidal functions.

Problem 6: Evaluate the integral

Problem 7: Evaluate the integral

Problem 8: Evaluate the integral

Hint: use the result from Problem 4. You might use the identity $\sin 2x = 2\sin x \cos x$.

3. Solutions

Problem 1:

Problem 2:

Problem 3:

Problem 4: Perform the substitution $x = a + b - u$ to get

Then notice that the integral is definite, and $u$ is a dummy variable — by renaming it $x$, we complete the proof of the King Property.

Problem 5:

Problem 6: Perform the substitution $u = x^{-4}$ to get

which evaluates to $-(1 + x^{-4})^{1/4} + C$.

Problem 7:

Problem 8:

Conclusion

Congratulations on making it through this handout! The problems were tough, so give yourself a pat on the back if you managed to do them all — you can now compute many different types of integrals!